Trim off the ends of the chard stems to freshen them up. Cut the stems from the leaves, rinse them well (chard sometimes hides grit or mud, so pay attention), and then cut the stems crosswise into 1/4–inch slices.
Stack the leaves, roll into a cylinder, and cut crosswise into 1/2-inch ribbons. Put the sliced leaves in a colander and rinse thoroughly.
Put 2 tablespoons of the oil in a large skillet over medium-high heat. Add the chard stems, season with 1/2 teaspoon salt, and sauté until soft, fragrant, and just starting to turn golden, 6 to 8 minutes. While they cook, peel and mince the garlic. Add it to the greens, sauté for another 30 seconds (take care not to brown it), and then add the chard leaves. They may not all fit in the pan right away, but as they wilt, you can add more.
Cook over medium-high heat, tossing frequently until the chard is wilted and starting to get tender, 5 to 6 minutes. Add the arugula and cook until it fully wilts, 2 to 3 more minutes.
Transfer the greens to a colander to let any remaining moisture drain off, squeezing gently to get the greens as dry as possible.
Roughly chop the leaves and tender stems of the cilantro and parsley, as well as the mint and dill.
Add the herbs and toss – your hands will be the best tool – to incorporate. Add the feta, lemon zest, and hot sauce; toss again. Add pepper to taste and more salt, lemon, and/or hot sauce to taste—you want the greens to be quite savory.
Heat an oven to 375°F. Melt the butter and combine the remaining 6 tablespoons olive oil in a small bowl. Stack the phyllo sheets on the counter and cover them with a large sheet of plastic wrap or a clean dish towel.
Brush the interior of a 9-inch (23-cm) pie plate with a coating of the melted butter and oil.
Brush 1 sheet of phyllo with the mixture and drape it inside the pie plate, buttered side up. Brush a second sheet and drape it on top, but at a 45-degree angle to the first sheet. Brush one-third of the sheet and drape it in the opposite direction, so that the sheets form a sort of star pattern and the edges of the pie plate are covered.
Pile the chard-feta mixture into the pastry-lined pie plate. Brush the fourth sheet, fold it in half width-wise, and place it on top of the greens; brush the surface. Repeat with the next sheet, placing it perpendicular to the first to cover the filling. Lightly brush all the edges of the phyllo hanging over the edge of the plate, including the undersides (this will make them more supple
and easier to fold. Gently fold up the overhang to make an edge and then cut five slits in the top of the pie in a star pattern, to help steam escape.
Brush the remaining sheets generously and gently scrunch them and place on the surface of the pie to make a freeform ruffle. Brush everything again with any remaining oil-butter mixture, though try to avoid the milky liquid from the butter.
Bake until the pastry is deep golden and crisp, 35 to 40 minutes. Let the pie cool for about 20 minutes before cutting into wedges to serve.